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Theorem List for Metamath Proof Explorer - 27101-27200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremkonigsbergiedgw 27101* The indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}

TheoremkonigsbergiedgwOLD 27102* The indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) Obsolete version of konigsbergiedgw 27101 as of 9-Mar-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}

Theoremkonigsbergssiedgwpr 27103* Each subset of the indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V ∧ 𝐸 = (𝐴 ++ 𝐵)) → 𝐴 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremkonigsbergssiedgw 27104* Each subset of the indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V ∧ 𝐸 = (𝐴 ++ 𝐵)) → 𝐴 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremkonigsbergumgr 27105 The Königsberg graph 𝐺 is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐺 ∈ UMGraph

TheoremkonigsbergupgrOLD 27106 The Königsberg graph 𝐺 is a pseudograph. (Contributed by AV, 28-Feb-2021.) Obsolete version of konigsbergumgr 27105 as of 9-Mar-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐺 ∈ UPGraph

Theoremkonigsberglem1 27107 Lemma 1 for konigsberg 27112: Vertex 0 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((VtxDeg‘𝐺)‘0) = 3

Theoremkonigsberglem2 27108 Lemma 2 for konigsberg 27112: Vertex 1 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((VtxDeg‘𝐺)‘1) = 3

Theoremkonigsberglem3 27109 Lemma 3 for konigsberg 27112: Vertex 3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((VtxDeg‘𝐺)‘3) = 3

Theoremkonigsberglem4 27110* Lemma 4 for konigsberg 27112: Vertices 0, 1, 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       {0, 1, 3} ⊆ {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}

Theoremkonigsberglem5 27111* Lemma 5 for konigsberg 27112: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})

Theoremkonigsberg 27112 The Königsberg Bridge problem. If 𝐺 is the Königsberg graph, i.e. a graph on four vertices 0, 1, 2, 3, with edges {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 2}, {2, 3}, {2, 3}, then vertices 0, 1, 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eulerpath 27094 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (EulerPaths‘𝐺) = ∅

16.5  The Friendship Theorem

16.5.1  Friendship graphs - basics

Syntaxcfrgr 27113 Extend class notation with friendship graphs.
class FriendGraph

Definitiondf-frgr 27114* Define the class of all friendship graphs: a simple graph is called a friendship graph if every pair of its vertices has exactly one common neighbor. This condition is called the friendship condition , see definition in [MertziosUnger] p. 152. (Contributed by Alexander van der Vekens and Mario Carneiro, 2-Oct-2017.) (Revised by AV, 29-Mar-2021.)
FriendGraph = {𝑔 ∣ (𝑔 ∈ USGraph ∧ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒)}

Theoremisfrgr 27115* The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))

Theoremfrgrusgrfrcond 27116* A friendship graph is a simple graph which fulfils the friendship condition. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))

Theoremfrgrusgr 27117 A friendship graph is a simple graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
(𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )

Theoremfrgr0v 27118 Any null graph (set with no vertices) is a friendship graph iff its edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ FriendGraph ↔ (iEdg‘𝐺) = ∅))

Theoremfrgr0vb 27119 Any null graph (without vertices and edges) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 29-Mar-2021.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ FriendGraph )

Theoremfrgruhgr0v 27120 Any null graph (without vertices) represented as hypergraph is a friendship graph. (Contributed by AV, 29-Mar-2021.)
((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ FriendGraph )

Theoremfrgr0 27121 The null graph (graph without vertices) is a friendship graph. (Contributed by AV, 29-Mar-2021.)
∅ ∈ FriendGraph

Theoremrspc2vd 27122* Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class 𝐷 for the second set variable 𝑦 may depend on the first set variable 𝑥. (Contributed by AV, 29-Mar-2021.)
(𝑥 = 𝐴 → (𝜃𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐸)    &   (𝜑𝐵𝐸)       (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃𝜓))

Theoremfrcond1 27123* The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))

Theoremfrcond2 27124* The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))

Theoremfrgreu 27125* Variant of frcond2 27124: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))

Theoremfrcond3 27126* The friendship condition, expressed by neighborhoods: in a friendship graph, the neighborhood of a vertex and the neighborhood of a second, different vertex have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))

Theoremfrcond4 27127* The friendship condition, alternatively expressed by neighborhoods: in a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})

16.5.2  The friendship theorem for small graphs

Theoremfrgr1v 27128 Any graph with (at most) one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ FriendGraph )

Theoremnfrgr2v 27129 Any graph with two (different) vertices is not a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Proof shortened by Alexander van der Vekens, 13-Sep-2018.) (Revised by AV, 29-Mar-2021.)
(((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph )

Theoremfrgr3vlem1 27130* Lemma 1 for frgr3v 27132. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) → ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))

Theoremfrgr3vlem2 27131* Lemma 2 for frgr3v 27132. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))

Theoremfrgr3v 27132 Any graph with three vertices which are completely connected with each other is a friendship graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Revised by AV, 29-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph ) → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))))

Theorem1vwmgr 27133* Every graph with one vertex (which may be connect with itself by (multiple) loops!) is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Revised by AV, 31-Mar-2021.)
((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸))

Theorem3vfriswmgrlem 27134* Lemma for 3vfriswmgr 27135. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))

Theorem3vfriswmgr 27135* Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))

Theorem1to2vfriswmgr 27136* Every friendship graph with one or two vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))

Theorem1to3vfriswmgr 27137* Every friendship graph with one, two or three vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ 𝐸)))

Theorem1to3vfriendship 27138* The friendship theorem for small graphs: In every friendship graph with one, two or three vertices, there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))

16.5.3  Theorems according to Mertzios and Unger

Theorem2pthfrgrrn 27139* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 15-Nov-2017.) (Revised by AV, 1-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))

Theorem2pthfrgrrn2 27140* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 16-Nov-2017.) (Revised by AV, 1-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐)))

Theorem2pthfrgr 27141* Between any two (different) vertices in a friendship graph, tere is a 2-path (simple path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 1-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑓𝑝(𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑝 ∧ (#‘𝑓) = 2))

Theorem3cyclfrgrrn1 27142* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))

Theorem3cyclfrgrrn 27143* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))

Theorem3cyclfrgrrn2 27144* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))

Theorem3cyclfrgr 27145* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑣𝑉𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣))

Theorem4cycl2v2nb 27146 In a (maybe degenerated) 4-cycle, two vertice have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))

Theorem4cycl2vnunb 27147* In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)

Theoremn4cyclfrgr 27148 There is no 4-cycle in a friendship graph, see Proposition 1(a) of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
((𝐺 ∈ FriendGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (#‘𝐹) ≠ 4)

Theorem4cyclusnfrgr 27149 A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → 𝐺 ∉ FriendGraph ))

Theoremfrgrnbnb 27150 If two neighbors 𝑈 and 𝑊 of a vertex 𝑋 have a common neighbor 𝐴 in a friendship graph, then this common neighbor 𝐴 must be the vertex 𝑋. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 2-Apr-2021.)
𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)       ((𝐺 ∈ FriendGraph ∧ (𝑈𝐷𝑊𝐷) ∧ 𝑈𝑊) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋))

Theoremfrgrconngr 27151 A friendship graph is connected, see remark 1 in [MertziosUnger] p. 153 (after Proposition 1): "An arbitrary friendship graph has to be connected, ... ". (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 1-Apr-2021.)
(𝐺 ∈ FriendGraph → 𝐺 ∈ ConnGraph)

Theoremvdgn0frgrv2 27152 A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0))

Theoremvdgn1frgrv2 27153 Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))

Theoremvdgn1frgrv3 27154* Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 4-Sep-2018.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)

Theoremvdgfrgrgt2 27155 Any vertex in a friendship graph (with more than one vertex - then, actually, the graph must have at least three vertices, because otherwise, it would not be a friendship graph) has at least degree 2, see remark 3 in [MertziosUnger] p. 153 (after Proposition 1): "It follows that deg(v) >= 2 for every node v of a friendship graph". (Contributed by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 5-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → 2 ≤ ((VtxDeg‘𝐺)‘𝑁)))

16.5.4  Huneke's Proof of the Friendship Theorem

In this section, the friendship theorem friendship 27241 is proven by formalizing Huneke's proof, see [Huneke] pp. 1-2. The three claims (see frgrncvvdeq 27166, frgrregorufr 27176 and frrusgrord0 27191) and additional statements (numbered in the order of their occurence in the paper) in Huneke's proof are cited in the corresponding theorems.

Theoremfrgrncvvdeqlem1 27156 Lemma 1 for frgrncvvdeq 27166. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 28-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝑋𝑁)

Theoremfrgrncvvdeqlem2 27157* Lemma 2 for frgrncvvdeq 27166. In a friendship graph, for each neighbor of a vertex there is exactly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)

Theoremfrgrncvvdeqlem3 27158* Lemma 3 for frgrncvvdeq 27166. The unique neighbor of a vertex (expressed by a restricted iota) is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

Theoremfrgrncvvdeqlem4 27159* Lemma 4 for frgrncvvdeq 27166. The mapping of neighbors to neighbors is a function. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝐴:𝐷𝑁)

Theoremfrgrncvvdeqlem5 27160* Lemma 5 for frgrncvvdeq 27166. The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

Theoremfrgrncvvdeqlem6 27161* Lemma 6 for frgrncvvdeq 27166. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸)

Theoremfrgrncvvdeqlem7 27162* Lemma 7 for frgrncvvdeq 27166. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)

Theoremfrgrncvvdeqlem8 27163* Lemma 8 for frgrncvvdeq 27166. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Revised by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝐴:𝐷1-1𝑁)

Theoremfrgrncvvdeqlem9 27164* Lemma 9 for frgrncvvdeq 27166. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝐴:𝐷onto𝑁)

Theoremfrgrncvvdeqlem10 27165* Lemma 10 for frgrncvvdeq 27166. (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (𝐺 NeighbVtx 𝑋)    &   𝑁 = (𝐺 NeighbVtx 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝐺 ∈ FriendGraph )    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))       (𝜑𝐴:𝐷1-1-onto𝑁)

Theoremfrgrncvvdeq 27166* In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))

Theoremfrgrwopreglem1 27167* Lemma 1 for frgrwopreg 27172: the classes 𝐴 and 𝐵 are sets. The definition of 𝐴 and 𝐵 corresponds to definition 3 in [Huneke] p. 2: "Let A be the set of all vertices of degree k, let B be the set of all vertices of degree different from k, ..." (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       (𝐴 ∈ V ∧ 𝐵 ∈ V)

Theoremfrgrwopreglem2 27168* Lemma 2 for frgrwopreg 27172. If the set 𝐴 of vertices of degree 𝐾 is not empty in a friendship graph with at least two vertices, then 𝐾 must be greater than 1 . This is only an observation, which is not required for the proof the friendship theorem. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 2-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉) ∧ 𝐴 ≠ ∅) → 2 ≤ 𝐾)

Theoremfrgrwopreglem3 27169* Lemma 3 for frgrwopreg 27172. The vertices in the sets 𝐴 and 𝐵 have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 2-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))

Theoremfrgrwopreglem4 27170* Lemma 4 for frgrwopreg 27172. In a friendship graph each vertex with degree 𝐾 is connected with a vertex with degree other than 𝐾. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 2-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ 𝐸)

Theoremfrgrwopreglem5 27171* Lemma 5 for frgrwopreg 27172. If 𝐴 as well as 𝐵 contain at least two vertices, there is a 4-cycle in a friendship graph. This corresponds to statement 6 in [Huneke] p. 2: "... otherwise, there are two different vertices in A, and they have two common neighbors in B, ...". (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 3-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))

Theoremfrgrwopreg 27172* In a friendship graph there are either no vertices (𝐴 = ∅) or exactly one vertex ((#‘𝐴) = 1) having degree 𝐾, or all (𝐵 = ∅) or all except one vertices ((#‘𝐵) = 1) have degree 𝐾. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 3-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))

Theoremfrgrwopreg1 27173* According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised by AV, 11-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (#‘𝐴) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)

Theoremfrgrwopreg2 27174* According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised by AV, 11-May-2021.) (Proof shortened by AV, 3-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (#‘𝐵) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)

Theoremfrgrregorufr0 27175* In a friendship graph there are either no vertices having degree 𝐾, or all vertices have degree 𝐾 for any (nonnegative integer) 𝐾, unless there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "... all vertices have degree k, unless there is a universal friend." (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised by AV, 11-May-2021.) (Proof shortened by AV, 3-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))

Theoremfrgrregorufr 27176* If there is a vertex having degree 𝐾 for each (nonnegative integer) 𝐾 in a friendship graph, then either all vertices have degree 𝐾 or there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       (𝐺 ∈ FriendGraph → (∃𝑎𝑉 (𝐷𝑎) = 𝐾 → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))

Theoremfrgrregorufrg 27177* If there is a vertex having degree 𝑘 for each nonnegative integer 𝑘 in a friendship graph, then there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". Variant of frgrregorufr 27176 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))

Theoremfrgr2wwlkeu 27178* For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑐𝑉 ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))

Theoremfrgr2wwlkn0 27179 In a friendship graph, there is always a path/walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)

Theoremfrgr2wwlk1 27180 In a friendship graph, there is exactly one walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 4-Jan-2022.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (#‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)

Theoremfrgr2wsp1 27181 In a friendship graph, there is exactly one simple path of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 13-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (#‘(𝐴(2 WSPathsNOn 𝐺)𝐵)) = 1)

Theoremfrgr2wwlkeqm 27182 If there is a (simple) path of length 2 from one vertex to another vertex and a (simple) path of length 2 from the other vertex back to the first vertex in a friendship graph, then the middle vertex is the same. This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 20-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 7-Jan-2022.)
((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

Theoremfrgrhash2wsp 27183 The number of simple paths of length 2 is n*(n-1) in a friendship graph with n vertices. This corresponds to the proof of claim 3 in [Huneke] p. 2: "... the paths of length two in G: by assumption there are ( n 2 ) such paths.". However, Huneke counts undirected paths, so obtains the result ((𝑛C2) = ((𝑛 · (𝑛 − 1)) / 2)), whereas we count directed paths, obtaining twice that number. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 10-Jan-2022.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (#‘(2 WSPathsN 𝐺)) = ((#‘𝑉) · ((#‘𝑉) − 1)))

Theoremfusgreg2wsplem 27184* Lemma for fusgreg2wsp 27187 and related theorems. (Contributed by AV, 8-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       (𝑁𝑉 → (𝑝 ∈ (𝑀𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))

Theoremfusgr2wsp2nb 27185* The set of paths of length 2 with a given vertex in the middle for a finite simple graph is the union of all paths of length 2 from one neighbor to another neighbor of this vertex via this vertex. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 8-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑀𝑁) = 𝑥 ∈ (𝐺 NeighbVtx 𝑁) 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩})

Theoremfusgreghash2wspv 27186* According to statement 7 in [Huneke] p. 2: "For each vertex v, there are exactly ( k 2 ) paths with length two having v in the middle, ..." in a finite k-regular graph. For directed simple paths of length 2 represented by length 3 strings, we have again k*(k-1) such paths, see also comment of frgrhash2wsp 27183. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 10-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))

Theoremfusgreg2wsp 27187* In a finite simple graph, the set of all paths of length 2 is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = 𝑥𝑉 (𝑀𝑥))

Theorem2wspmdisj 27188* The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       Disj 𝑥𝑉 (𝑀𝑥)

Theoremfusgreghash2wsp 27189* In a finite k-regular graph with N vertices there are N times "k choose 2" paths with length 2, according to statement 8 in [Huneke] p. 2: "... giving n * ( k 2 ) total paths of length two.", if the direction of traversing the path is not respected. For simple paths of length 2 represented by length 3 strings, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 19-May-2021.) (Proof shortened by AV, 12-Jan-2022.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(2 WSPathsN 𝐺)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1)))))

Theoremfrrusgrord0lem 27190* Lemma for frrusgrord0 27191. (Contributed by AV, 12-Jan-2022.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 ∈ ℂ ∧ (#‘𝑉) ∈ ℂ ∧ (#‘𝑉) ≠ 0))

Theoremfrrusgrord0 27191* If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))

Theoremfrrusgrord 27192 If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". Variant of frrusgrord0 27191, using the definition RegUSGraph (df-rusgr 26448). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.)
𝑉 = (Vtx‘𝐺)       ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))

Theoremnumclwlk3lem3 27193 Lemma 3 for numclwwlk3 27227. (Contributed by Alexander van der Vekens, 26-Aug-2018.)
((𝐾 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ 𝑁 ∈ (ℤ‘2)) → (((𝐾↑(𝑁 − 2)) − 𝑌) + (𝐾 · 𝑌)) = (((𝐾 − 1) · 𝑌) + (𝐾↑(𝑁 − 2))))

Theoremextwwlkfablem2lem 27194 Lemma for extwwlkfablem2 27197. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 𝑁𝑁 ∈ (ℤ‘2)) → (#‘(𝑤 substr ⟨0, (𝑁 − 2)⟩)) = (𝑁 − 2))

Theoremextwwlkfablem1 27195 Lemma 1 for extwwlkfab 27207. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 27-May-2021.)
(((𝐺 ∈ USGraph ∧ 𝑁 ∈ (ℤ‘2)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘(𝑁 − 2)) = (𝑊‘0)) → (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0)))

Theoremclwwlkextfrlem1 27196 Lemma for numclwwlk2lem1 27219. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-May-2021.)
(((𝑁 ∈ ℕ0𝑍 ∈ (Vtx‘𝐺)) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑊) ≠ (𝑊‘0))) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ (𝑊‘0)))

Theoremextwwlkfablem2 27197 Lemma 2 for extwwlkfab 27207. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 28-May-2021.)
(((𝐺 ∈ USGraph ∧ 𝑁 ∈ (ℤ‘3)) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0)) → (𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ ((𝑁 − 2) ClWWalksN 𝐺))

Theoremnumclwwlkovf 27198* Value of operation 𝐹, mapping a vertex 𝑣 and a positive integer 𝑛 to the "(For a fixed vertex v, let f(n) be the number of) walks from v to v of length n" according to definition 5 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 28-May-2021.)
𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})       ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})

Theoremnumclwwlkffin 27199* In a finite graph, the value of operation 𝐹 is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 28-May-2021.)
𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝑉 = (Vtx‘𝐺)       ((𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐹𝑁) ∈ Fin)

Theoremnumclwwlkffin0 27200* In a finite graph, the value of operation 𝐹 is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 2-Jun-2021.)
𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝑉 = (Vtx‘𝐺)       ((𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝐹𝑁) ∈ Fin)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
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